Product Decompositions of Groups

University Of Chicago, Chicago IL

Investigators

Abstract

The main aspect of the project is to consider decomposing families of finite and infinite groups as the product of cyclic, Abelian or solvable subgroups. The historical background comes from Ito's theorem, Lazard's work on analytic pro-p groups and the connection between bounded generation and the congruence subgroup property for arithmetic groups established by Lubotzky, Platonov and Rapinchuk. While the PI considers problems motivated by these major results, he also investigates cyclic, Abelian and solvable decompositions in a broader context: for finite simple groups, infinite permutation groups, geometric groups and groups acting on trees. Besides rather surprising constructions (e.g., the infinite symmetric group is the product of finitely many Abelian subgroups), the existence of such a decomposition leads to strong asymptotic restrictions (e.g. on subgroup growth). On the other hand, trying to show that a certain decomposition does not exist forces one to look at the subgroup structure from a new point of view which then can be analyzed further. An example for this is Hausdorff dimension of groups acting on rooted trees. The set of symmetries of any object or structure forms a group. Thus group theory naturally comes into the picture whenever one needs to analyze symmetries (e.g. in quantum physics and chemistry or inside mathematics in geometry, number theory, coding theory and topology). The main goal of the proposed project is to find certain 'coordinate systems' in families of groups and so generate these groups in a transparent way using simple structures or to show that such nice systems do not exist. An elementary example is the fact that every plane isometry can be obtained as the composition of at most three reflections but not as the composition of two. There are manifold connections between the existence of such 'coordinate systems' and various other interesting group properties. Other than deep and useful results, the project also has elementary aspects which can be presented on an undergraduate level and thus serve educational purposes.